A note on exponents vs root heights for complex simple Lie algebras
Sankaran Viswanath
Department of Mathematics University of California
Davis, CA 95616, USA [email protected]
Submitted: Sep 8, 2006; Accepted: Nov 26, 2006; Published: Dec 7, 2006 Mathematics Subject Classification: 05E15
Abstract
We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebrag, the partition formed by the exponents ofg is dual to that formed by the numbers of positive roots at each height.
Let g be a finite dimensional, complex simple Lie algebra of rank n with associated root system ∆, simple roots αi (i = 1· · ·n) and set of positive roots ∆+. Let Q be the root lattice of gandQ+ denote the set comprising Z≥0 linear combinations of the αi. For each α ∈ Q, let eα denote the corresponding formal exponential; these satisfy the usual rules: e0 = 1 and eα+β = eαeβ. We define A := Q[t] [[e−α1,· · · , e−αn]]. Thus a typical element of A is a power series of the form P
β∈Q+cβ(t)e−β where each cβ(t)∈Q[t].
Consider the element ξ∈ A defined by:
ξ:= Y
α∈∆+
1−e−α 1−te−α
= Y
α∈∆+
(1 + (t−1)e−α+t(t−1)e−2α+t2(t−1)e−3α+· · ·) (1) Given β =Pn
i=1biαi ∈Q+, define its height to be htβ :=
Xn
i=1
bi
The main objective of this short note is to give an elementary combinatorial proof of the following proposition:
Proposition 1 Forβ ∈∆+, the coefficient of e−β in ξ is (tht(β)−tht(β)−1)
This proposition is theq = 0 case of a more general (q, t) theorem obtained by Bazlov [1] and Ion [2]. They consider
K˜(q, t) = Y
α∈∆+
Y
i≥0
(1−qie−α)(1−qi+1eα) (1−tqie−α)(1−tqi+1eα)
If [ ˜K(q, t)] denotes the constant term (coefficient ofe0) of ˜K(q, t), one defines ˜C(q, t) :=
K(q, t)/[ ˜˜ K(q, t)] (upto a minor difference in convention, this is called the Cherednik kernel in [2] ). Bazlov and Ion compute the coefficient of e−β inC(q, t) for β a positive root of g. Their approaches use techniques from Cherednik’s theory of Macdonald polynomials.
When q = 0, ˜C(0, t) reduces to ξ introduced above. Though proposition 1 is only a special case, it has a very interesting consequence. Ion showed [2] that it can be used to give a quick and elegant proof of the classical fact that for a finite dimensional simple Lie algebrag, the partition formed by listing its exponents in descending order is dual to the partition formed by the numbers of positive roots at each height (see below). This fact, first observed empirically by Shapiro and Steinberg was later proved by Kostant [3] using his theory of principal three dimensional subalgebras of g and by Macdonald [4] via his factorization of the Poincar´e series of the Weyl group of g.
The motivation for our approach to proposition 1 is to thereby obtain a proof of this classical fact via elementary means (bypassing Macdonald-Cherednik theory).
For completeness sake, we first quickly recall [2] how one can use proposition 1 to deduce the classical fact concerning exponents and heights of roots.
Let P denote the weight lattice of g and W its Weyl group. For our definition of exponents, we use the Kostka-Foulkes polynomialKα,0˜ (t) where ˜αis the highest long root of g. It is well known that this is given by
Kα,0˜ (t) = Xl
j=1
tmj (2)
where m1, m2,· · ·, ml are the exponents of g. The Kostka-Foulkes polynomials are the elements of the transition matrix between the Schur and the Hall-Littlewood bases of Q[t][P]W. They may be alternatively defined via Lusztig’st−analog of weight multiplicity;
we have
Kα,0˜ (t) = X
w∈W
(−1)`(w)P(w( ˜α+ρ)−(0 +ρ);t)
= coeff. of e0 in X
w∈W
(−1)`(w)ew( ˜α+ρ)−ρ Y (1−te−α)
where P(;t) is thet−analog of Kostant’s partition function.
The last equation can be rewritten as :
Kα,0˜ (t) = coeff. of e0 in X
w∈W
(−1)`(w)ew( ˜α+ρ)−ρ Y
α∈∆+
(1−e−α) ·ξ (3)
where ξ was defined earlier. The expression in (3) (from which we need to extract the coefficient of e0) is just the productχα˜ξ where χα˜ is the formal character of the adjoint representation ofg. This follows from the Weyl character formula and the fact that adjoint representation is irreducible with highest weight ˜α. Now,
1. χα˜ =le0+ X
α∈∆+
(eα+e−α) and
2. From equation (1), the power series for ξ has constant term 1 and only involves terms of the forme−γ for γ ∈Q+.
Thus,
coeff. ofe0 in χα˜ξ=l+ X
α∈∆+
( coeff. ofe−α inξ) From proposition 1, the right hand side equals l+ X
α∈∆+
(tht(α)−tht(α)−1). Letting ai :=
#{β∈∆+ : htβ =i}, this last sum becomes l+X
i≥1
ai(ti−ti−1) = (a1−a2)t+ (a2−a3)t2+· · · (sincea1 =l). Comparing with equation (2), we get ai−ai+1 is the number of timesi appears as an exponent ofg. This is exactly the classical result.
Proof of proposition 1: Givenγ ∈Q+, let Par(γ) be the set of all partitions ofγ into a sum of positive roots. Given such a partition π ∈Par(γ), say
π: γ = X
α∈∆+
cαα (cα ∈Z≥0)
let n(π) := X
α
cα be the total number of parts (counting repetitions) and d(π) := #{α: cα6= 0} be the number of distinct parts inπ. From equation (1), it is clear that
Coeff. ofe−γ in ξ= X
π∈Par(γ)
tn(π)−d(π)(t−1)d(π) (4)
Notation:
1. Given a subset A⊂Par(γ), let wt(A) :=X
π∈A
tn(π)−d(π)(t−1)d(π). Thus the coeff. of e−γ inξ equals wt(Par(γ)).
2. Given a simple root αi, let
Par(γ, αi) := {π∈Par(γ) :αi occurs as one of the parts inπ}
Par(γ,αbi) := {π∈Par(γ) :αi does not occur as a part in π}
Let (·,·) denote a nondegenerate,W−invariant symmetric bilinear form on the dual of the Cartan subalgebra and let sj ∈W (j = 1· · ·n) be the simple reflection correponding toαj.
We will prove proposition 1 by induction on htβ. If htβ = 1, β is a simple root. It is then clear from equation (1) that the coefficient of e−β is t−1 = t1 −t0. Now suppose β ∈∆+ with h:= htβ ≥2. Assume the proposition is true for all positive roots of height
< h. Choose a simple root αi such that (β, αi)>0 (such αi exists since (β, β)>0). Now h≥2 implies that siβ =β−2(α(β,αi)
i,αi)αi is a positive root of height< h.
Fact 1: Par(β,αbi) is in bijection with Par(siβ,αbi) Proof: Given a partition π : β = X
α∈∆+
cαα ∈ Par(β,αbi), we can form a partition of siβ as follows:
˜
π: siβ = X
α∈∆+
cα(siα)
Since αi is not one of the parts of π, all the parts of ˜π are positive roots, none equal to αi. It is clear that π7→ π˜ sets up the required bijection. Further, since n(π) =n(˜π) and d(π) =d(˜π), we have
wt(Par(β,αbi)) = wt(Par(siβ,αbi)) (5) Fact 2:
wt(Par(β, αi)) =twt(Par(β−αi, αi)) + (t−1) wt(Par(β−αi,αbi)) (6) Proof: There is an obvious bijection between the sets Par(β−αi) and Par(β, αi) obtained by sending a partitionπ in the first set to the partition ¯π obtained by adjoining the extra part αi to π. In order to see how wt(π) compares with wt(¯π), we write Par(β −αi) = Par(β−αi, αi)∪Par(β−αi,αbi). Forπ∈Par(β−αi, αi), the extra partαi in ¯π is a repeat part and thus
wt(¯π) =t wt(π)
while for π ∈ Par(β−αi,αbi), the extra αi in ¯π is a new distinct part and thus wt(¯π) = (t−1) wt(π). This proves equation (6).
Let k:= 2(α(β,αi)
i,αi) >0 and consider the αi−string through β:
β, β−αi,· · · , β−kαi
Each of these is a positive root. We now rewrite equation (6) as
wt(Par(β, αi))−wt(Par(β−αi, αi)) = (t−1) wt(Par(β−αi))
Iterating this equation k times with β −jαi in place of β (0≤ j ≤k−1) and summing the resulting equations, we get
wt(Par(β, αi))−wt(Par(β−kαi, αi)) = (t−1) Xk
j=1
wt(Par(β−jαi)) By induction hypothesis,
wt(Par(β−jαi)) =th−j−th−j−1 Further,
wt(Par(β−kαi, αi)) = wt(Par(β−kαi))−wt(Par(β−kαi,αbi))
= (th−k−th−k−1)−wt(Par(β−kαi,αbi)) Since β−kαi =siβ, we can use equation (5). This gives
wt(Par(β, αi)) + wt(Par(β,αbi)) = (t−1) Xk
j=1
(th−j −th−j−1) + (th−k−th−k−1)
=th−th−1
Since the left hand side equals wt(Par(β)), proposition 1 is proved.
Acknowledgements: The author would like to thank John Stembridge for bringing refer- ences [1] and [2] to his attention and for his comments on an earlier draft of this note.
References
[1] Yuri Bazlov. Graded multiplicities in the exterior algebra.Adv. Math., 158(2):129–153, 2001.
[2] Bogdan Ion. The Cherednik kernel and generalized exponents. Int. Math. Res. Not., (36):1869–1895, 2004.
[3] Bertram Kostant. The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math., 81:973–1032, 1959.
[4] I. G. Macdonald. The Poincar´e series of a Coxeter group. Math. Ann., 199:161–174, 1972.
